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The Measurement of Subjective Probability
Published online by Cambridge University Press: 13 April 2024
Summary
Beliefs come in degrees, and we often represent those degrees with numbers. We might say, for example, that we are 90% confident in the truth of some scientific hypothesis, or only 30% confident in the success of some risky endeavour. But what do these numbers mean? What, in other words, is the underlying psychological reality to which the numbers correspond? And what constitutes a meaningful difference between numerically distinct representations of belief? In this Element, we discuss the main approaches to the measurement of belief. These fall into two broad categories-epistemic and decision-theoretic-with divergent foundations in the theory of measurement. Epistemic approaches explain the measurement of belief by appeal to relations between belief states themselves, whereas decision-theoretic approaches appeal to relations between beliefs and desires in the production of choice and preferences.
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- Online ISBN: 9781009401319Publisher: Cambridge University PressPrint publication: 02 May 2024
References
Alon, S. and Lehrer, E. (2014). Subjective multi-prior probability: A representation of a partial likelihood relation. Journal of Economic Theory 151(C), 476–92.CrossRefGoogle Scholar
Alon, S. and Schmeidler, D. (2014). Purely subjective maxmin expected utility. Journal of Economic Theory 152, 382–412.CrossRefGoogle Scholar
Augustin, T., Coolen, F., Cooman, G., and Troffaes, M. (Eds.) (2014). Introduction to Imprecise Probabilities. Wiley.CrossRefGoogle Scholar
Baccelli, J. (2020). Beyond the metrological viewpoint. Studies in History and Philosophy of Science Part A 80, 56–61.CrossRefGoogle ScholarPubMed
Bolker, E. (1967). A simultaneous axiomatization of utility and subjective probability. Philosophy of Science 34(4), 333–40.CrossRefGoogle Scholar
Borsboom, D. (2005). Measuring the Mind: Conceptual Issues in Contemporary Psychometrics. Cambridge University Press.CrossRefGoogle Scholar
Brickhill, H. and Horsten, L. (2018). Triangulating non-Archimedean probability. Review of Symbolic Logic 11(3), 519–46.CrossRefGoogle Scholar
Builes, D., Horowitz, S., and Schoenfield, M. (2022). Dilating and contracting arbitrarily. Nous 56(1), 3–20.CrossRefGoogle Scholar
Bunge, M. (1973). On confusing ‘measure’ with ‘measurement’ in the methodology of behavioral science. In The Methodological Unity of Science, pp. 105–22. D. Reidel Publishing.CrossRefGoogle Scholar
Chalmers, D. (2011). The nature of epistemic space. In Egan, A. and Weatherson, B. (Eds.), Epistemic Modality, pp. 60–107. Oxford University Press.CrossRefGoogle Scholar
Christensen, D. (2001). Preference-based arguments for probabilism. Philosophy of Science 68(3), 356–76.CrossRefGoogle Scholar
Clark, S. (2000). The measurement of qualitative probability. Journal of Mathematical Psychology 44(3), 464–79.CrossRefGoogle ScholarPubMed
Davidson, D. and Suppes, P. (1956). A finitistic axiomatization of subjective probability and utility. Econometrica 24(3), 264–75.CrossRefGoogle Scholar
Davidson, D., Suppes, P., and Siegel, S. (1957). Decision Making: An Experimental Approach. Stanford University Press.Google Scholar
de Finetti, B. (1931). Sul significato soggettivo della probabilita. Fundamenta Mathematicae 17(1), 298–329.CrossRefGoogle Scholar
Debreu, G. (1959). Cardinal utility for even-chance mixtures of pairs of sure prospects. The Review of Economic Studies 28(3), 174–7.Google Scholar
Decoene, S., Onghena, P., and Janssen, R. (1995). Representationalism under attack: Review of an introduction to the logic of psychological measurement. Journal of Mathematical Psychology 39(2), 234–42.CrossRefGoogle Scholar
DiBella, N. (2018). The qualitative paradox of non-conglomerability. Synthese 195(3), 1181–210.CrossRefGoogle Scholar
Domotor, Z. (1970). Qualitative information and entropy structures. In Hintikka, J. and Suppes, P. (Eds.), Information and Inference, pp. 148–94. Reidel.Google Scholar
Domotor, Z. (1978). Axiomatization of Jeffrey utilities. Synthese 39(2), 165–210.CrossRefGoogle Scholar
Elliott, E. (2017a). Probabilism, representation theorems, and whether deliberation crowds out prediction. Erkenntnis 82(2), 379–99.CrossRefGoogle Scholar
Elliott, E. (2017b). Ramsey without ethical neutrality: A new representation theorem. Mind 126(501), 1–51.Google Scholar
Elliott, E. (2017c). A representation theorem for frequently irrational agents. Journal of Philosophical Logic 46(5), 467–506.CrossRefGoogle Scholar
Elliott, E. (2019b). Impossible worlds and partial belief. Synthese 196(8), 3433–58.CrossRefGoogle Scholar
Eriksson, L. and Hájek, A. (2007). What are degrees of belief? Studia Logica 86(2), 183–213.CrossRefGoogle Scholar
Evren, O. and Ok, E. (2011). On the multi-utility representation of preference relations. Journal of Mathematical Economics 47(4–5), 554–63.CrossRefGoogle Scholar
Fine, T. (1973). Theories of Probability: An Examination of Foundations. Academic Press.Google Scholar
Fishburn, P. (1967). Preference-based definitions of subjective probability. The Annals of Mathematical Statistics 38(6), 1605–17.CrossRefGoogle Scholar
Hájek, A. (2003). What conditional probability could not be. Synthese 137(3), 273–323.CrossRefGoogle Scholar
Halpern, J. (2001). Lexicographic probability, conditional probability, and nonstandard probability. In Proceedings of the 8th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 17–30. Morgan Kaufmann Publishers.Google Scholar
Hawthorne, J. (2016). A logic of comparative support: Qualitative conditional probability relations representable by Popper functions. In Hájek, A. and Hitchcock, C. (Eds.), Oxford Handbook of Probabilities and Philosophy, pp. 277–95. Oxford University Press.Google Scholar
Hölder, O. (1901). Die Axiome der Quantitat und die Lehre vom Mass. Berichte über die Verhandlungen der Königlich-Sáchsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 53, 1–63.Google Scholar
Jeffrey, R. (1968). Probable knowledge. Studies in Logic and the Foundations of Mathematics 51, 166–90.CrossRefGoogle Scholar
Jeffrey, R. (1978). Axiomatizing the logic of decision. In Foundations and Applications of Decision Theory, pp. 227–31. Springer.Google Scholar
Jeffrey, R. (1990). The Logic of Decision (Second Edition). University of Chicago Press.Google Scholar
Joyce, J. (2010). A defense of imprecise credences in inference and decision making. Philosophical Perspectives 24(1), 281–323.CrossRefGoogle Scholar
Kahneman, D. and Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica 47(2), 263–91.CrossRefGoogle Scholar
Kaplan, M. and Fine, T. (1977). Joint orders in comparative probability. The Annals of Probability 5(2), 161–79.CrossRefGoogle Scholar
Koopman, B. (1940a). The axioms and algebra of intuitive probability. Annals of Mathematics 41(2), 269–92.CrossRefGoogle Scholar
Koopman, B. (1940b). The bases of probability. Bulletin of the American Mathematical Society 46(10), 763–74.CrossRefGoogle Scholar
Kraft, C., Pratt, J., and Seidenberg, A. (1959). Intuitive probability on finite sets. The Annals of Mathematical Statistics 30(2), 408–19.CrossRefGoogle Scholar
Krantz, D., Luce, R., Suppes, P., and Tversky, A. (1971). Foundations of Measurement, Vol. I: Additive and Polynomial Representations. Academic Press.Google Scholar
Lewis, D. (1979). Attitudes de dicto and de se. The Philosophical Review 88(4), 513–43.CrossRefGoogle Scholar
Luce, D. R. (1968). On the numerical representation of qualitative conditional probability. The Annals of Mathematical Statistics 39(2), 481–91.CrossRefGoogle Scholar
Luce, R. (1978). Dimensionally invariant numerical laws correspond to meaningful qualitative relations. Philosophy of Science 45(1), 1–16.CrossRefGoogle Scholar
Luce, R., Krantz, D., Suppes, P., and Tversky, A. (1990). Foundations of Measurement, Vol. III: Representation, Axiomatization, and Invariance. Dover.Google Scholar
Luce, R. and Narens, L. (1978). Qualitative independence in probability theory. Theory and Decision 9(3), 225–39.CrossRefGoogle Scholar
Luce, R. and Tukey, J. (1964). Simultaneous conjoint measurement: A new scale type of fundamental measurement. Journal of Mathematical Psychology 1(1), 1–27.CrossRefGoogle Scholar
Mari, L. (2005). The problem of foundations of measurement. Measurement 38(4), 259–66.CrossRefGoogle Scholar
Mari, L., Carbone, P., Giordani, A., and Petri, D. (2017). A structural interpretation of measurement and some related epistemological issues. Studies in History and Philosophy of Science 65–66, 46–56.CrossRefGoogle ScholarPubMed
Mayo-Wilson, C. and Wheeler, G. (2019). Epistemic decision theory’s reckoning. Manuscript. http://philsci-archive.pitt.edu/16374/1/25a_EDTR.pdf.Google Scholar
Meacham, C., and Weisberg, J. (2011). Representation theorems and the foundations of decision theory. Australasian Journal of Philosophy 89(4), 641–63.CrossRefGoogle Scholar
Michell, J. (2021). Representational measurement theory: Is its number up? Theory & Psychology 31, 3–23.CrossRefGoogle Scholar
Mundy, B. (1987). The metaphysics of quantity. Philosophical Studies 51, 29–54.CrossRefGoogle Scholar
Mundy, B. (1994). Quantity, representation and geometry. In Humphries, P. (Ed.), Patrick Suppes: Scientific Philosopher, pp. 59–102. Kluwer.CrossRefGoogle Scholar
Narens, L. (1980). On qualitative axiomatizations for probability theory. Journal of Philosophical Logic 9, 143–51.CrossRefGoogle Scholar
Narens, L. (1981). On the scales of measurement. Journal of Mathematical Psychology 24(3), 249–75.CrossRefGoogle Scholar
Narens, L. (1985). Abstract Measurement Theory. Massachusetts Institute of Technology Press.Google Scholar
Narens, L., and Luce, D. (1993). Further comments on the ‘nonrevolution’ arising from axiomatic measurement theory. Psychological Science 4, 127–30.CrossRefGoogle Scholar
Nolan, D. (1997). Impossible worlds: A modest approach. Notre Dame Journal of Formal Logic 38, 535–72.CrossRefGoogle Scholar
Ramsey, F. (1931). Truth and probability. In Braithwaite, R. (Ed.), The Foundations of Mathematics and Other Logical Essays, pp. 156–98. London: Routledge.Google Scholar
Reiss, J. (2016). Error in Economics: Towards a More Evidence-Based Methodology. Routledge.CrossRefGoogle Scholar
Roberts, F. (1985). Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences. Cambridge University Press.Google Scholar
Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical Psychology 1(2), 233–47.CrossRefGoogle Scholar
Spohn, W. (1977). Where Luce and Krantz do really generalize Savage’s decision model. Erkenntnis 11(1), 113–34.CrossRefGoogle Scholar
Stalnaker, R. C. (1984). Inquiry. London: The Massachusetts Institute of Technology Press.Google Scholar
Stevens, S. (1946). On the theory of scales of measurement. Science 103(2684), 677–80.CrossRefGoogle ScholarPubMed
Suppes, P. (1969). Studies in the Methodology and Foundations of Science: Selected Papers from 1951 to 1969. Dordrecht: Springer.CrossRefGoogle Scholar
Suppes, P. (2014). Using Padoa’s principle to prove the non-definability, in terms of each other, of the three fundamental qualitative concepts of comparative probability, independence and comparative uncertainty, with some new axioms of qualitative independence and uncertainty included. Journal of Mathematical Psychology 60, 47–57.CrossRefGoogle Scholar
Suppes, P., and Pederson, A. (2016). Qualitative axioms of uncertainty as a foundation for probability and decision-making. Minds and Machines 26, 185–202.CrossRefGoogle Scholar
Suppes, P., and Zanotti, M. (1976). Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualitative probability ordering. Journal of Philosophical Logic 5(3), 431–8.CrossRefGoogle Scholar
Suppes, P., and Zanotti, M. (1982). Necessary and sufficient qualitative axioms for conditional probability. Z. Wahrschelnllchkeltstheorle verw. Gebiete 60, 163–9.Google Scholar
Suppes, P., and Zinnes, J. (1963). Basic measurement theory. In Luce, D. R. (Ed.), Handbook of Mathematical Psychology. John Wiley & Sons.Google Scholar
Swoyer, C. (1991). Structural representation and surrogative reasoning. Synthese 87(3), 449–508.CrossRefGoogle Scholar
Titelbaum, M. (2022). Fundamentals of Bayesian Epistemology 2: Arguments, Challenges, Alternatives. Oxford University Press.Google Scholar
van Fraassen, B. (1976). Representation of conditional probabilities. Journal of Philosophical Logic 5, 417–30.CrossRefGoogle Scholar
Wakker, P. (2004). On the composition of risk preference and belief. Psychological Review 111, 236–41.CrossRefGoogle Scholar
Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman & Hall.CrossRefGoogle Scholar
Zynda, L. (2000). Representation theorems and realism about degrees of belief. Philosophy of Science 67(1), 45–69.CrossRefGoogle Scholar
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